Digital Electronics

1. ELEC 256 / Saif Zahir
UBC / 2000
Sequential Logic Design
• Sequential Networks
Simple Circuits with Feedback
R-S Latch
J-K Flipflop
Edge -Triggered Flip-Flops
• Timing Methodologies
Cascading Flip-Flops for Proper Operation
Narrow Width Clocking vs. Multiphase Clocking
Clock Skew
• Realizing Circuits with Flip-Flops
Choosing a FF Type
Characteristic Equations
Conversion Among Types
• Self-Timed Circuits

2. ELEC 256 / Saif Zahir
UBC / 2000
Sequential Switching Networks
• Sequential logic forms basis for building
"memory" into circuits.
• Sequential logic is characterized by the
presence of feedback paths.
Combinational
Logic
Delay = D
x1
x2
x3
x4
z1
z2
z3
z4
z3 = F(x1, ... ,x4,z3,z4)
z3(t+D) = F(x1(t), ... ,x4(t),z3(t),z4(t))
Observations:
• z3 and z4 appear as both
inputs and outputs.
• The “state” of variable z3 (or z4) at
time t+D depends on its value at
time t, i.e. z3(t+D) = F(z3(t)),
hence, circuit has memory.
• z3(t) and z4(t) are called
state variables .
Sequential Circuit

3. ELEC 256 / Saif Zahir
UBC / 2000
Simple Sequential Circuits
Cascaded Inverters: Static Memory Cell "0"
"1"
Delay=D
x(t) z(t)
D
D
t
x
z
Assuming D > 0
z(t+D) = x(t) z(t)
if x(t) = 0 then z(t)=1 (stable state)
if x(t) = 1 then z(t+D) = z(t)
Another Example
Observe that
NAND gate with one input asserted
acts as an inverter with respect to
other input
When x=1, equaivalent circuit
z(t)
Timing Waveform:

4. ELEC 256 / Saif Zahir
UBC / 2000
Inverter Chains and Ring Oscillators
Inverter Chains
Odd # of stages leads to ring oscillator
Snapshot taken just before last inverter changes
Output high
propagating
thru this stage
Timing Waveform:
A (=X)
B
C
D
E
Period of Repeating Waveform (tp)
Gate Delay ( td)
0
1
0
1
0
1
tp = n D
n = no. inverters
A
B C D E
1 0 0
0
1
X

5. ELEC 256 / Saif Zahir
UBC / 2000
Cross-Coupled NOR Gates
Observation
NOR gate with one input=0, acts as an
inverter with respect to other input.
0
x
X
x(t)
z(t)
x=1 --> z=0
x=0 --> z=1
Problem: how can we insert x in the loop?
Simple-Latch: two-inverter loop
q
Q
R
S
Equivalent NOR circuit with two control
inputs (R and S) to break or close the loop
R: Reset input (R=1 --> Q=0)
S: Set input (S=1 --> Q=1)
q
Q
R
S
Alternative
representation

6. ELEC 256 / Saif Zahir
UBC / 2000
The RS Latch
q=0
Q=1
R=0
S=0
• if R=S=0 then Q(t+D)=Q(t) (memory element)
q=1
Q=0
R=0
S=0
q=0
Q=0
R=1
S=1
• if R=S=1 then q = Q = 0, which violates the inverter rule (q = 0, Q = 1)
• if R and S chnage from 1-to-0 at precisely same moment, then RS latch
will oscillate (provided the NOR gate delays are perfectly matched)
q=0-->1-->0-->1--
Q=0-->1-->0-->1--
R=1-->0
S=1-->0
0-->1-->0-->1
0-->1-->0-->1

7. ELEC 256 / Saif Zahir
UBC / 2000
State Behavior of RS Latch
Truth Table Summary
of R-S Latch Behavior
Q Q Q Q
Q Q
0 1 1 0
0 0
Q Q
1 1
Q
hold
0
1
unstable
S
0
0
1
1
R
0
1
0
1
The response and transient behavior of the RS latch can be described
using a state-diagram:
1- Nodes represent the unique states
of the circuit
2- Arcs indicate state-transition under
particular input combinations
(arc labels).
Because of the resulting unstable behavior
the combination R=S=1 is called the forbidden
input for the RS latch.
state 0
state 3
state 1 state 2

8. ELEC 256 / Saif Zahir
UBC / 2000
State-Diagrams and State Tables
Q Q Q Q
Q Q
0 1 1 0
0 0
SR = 1 0
SR = 0 1
SR = 0 1
SR = 1 1
SR = 1 0
SR = 1 1
SR = 00, 01 SR = 00, 10
Q Q
1 1
SR = 0 0
SR = 0 0, 11
SR = 11
SR = 1 0
SR = 0 1
qQ SR SR SR SR
00 01 10 11
00 11 01 10 00
01 01 01 10 00
10 10 01 10 00
00 00 01 10 00
PS NS (q+, Q+)
PS : present state
NS: next state
Q+ : Q(t+D)
A state-table expresses the same
information of the state-diagram
in a tabular format
Note the unstable behavior is now obvious from the continuous transition
states 00 and 11 when SR changes from 11 to 00.

9. ELEC 256 / Saif Zahir
UBC / 2000
The D-Latch
enabled when C=1
D
C
Clk
Enable
Q
q
if C=1 then Q=D
if C=0 then Q(t+D)=Q(t)
if C=0, then R=S=0
and Q(t+D)=Q(t)
If C=1 and D=0 then
R=1, S=0, and Q=0
if C=1 and D=1 then
R=0, S=1, and Q=1
Realization using an RS latch
Note that input R=S=1
can not occur
R
S Q
q
q
D
C
RS
Latch

10. ELEC 256 / Saif Zahir
UBC / 2000
Input
Clock
Tsu Th
Steup and Hold Times
Clock:
Periodic Event, causes state of memory
element to change.
There is a timing "window" around the
clocking event during which the input
must remain stable and unchanged
in order to be recognized
Setup Time (Tsu):
Minimum time before the clocking event by
which the input must be stable
Hold Time (Th)
Minimum time after the clocking event during
which the input must remain stable
Primitive Memory Elements:
Latches: Continuously sample their inputs. Any change in the level of the inputs
is propagated through to the outputs (level sensitive).
Flip-Flops: Outputs change only with respect to the clock, normally the rising edge
or the falling edges of the clock.

11. ELEC 256 / Saif Zahir
UBC / 2000
Level Sensitive Latches
S
R
Q
Q
enb
Timing Diagram:
Set Reset
RS latch with active-low inputs and active-low Enable
Truth Table
enb S R Q+
1 x x Q
0 0 0 Q
0 0 1 0
0 1 0 1
0 1 1 Unstable

12. ELEC 256 / Saif Zahir
UBC / 2000
Flip-Flops and Latches
7474
7476
Bubble here for negative
edge triggered device
Timing Diagram:
Behavior is the same unless input changes
occur while the clock is high
Edge triggered devices sample inputs on the rising
or falling edge of the Clock or the Enable.
Transparent latches sample inputs as long as the
clock is asserted -
output changes with input (after certain delay).
Positive edge-triggered
flip-flop
Level-sensitive
latch
D Q
D Q
C
Clk
Clk
D
Clk
Q
Q
7474
7476

13. ELEC 256 / Saif Zahir
UBC / 2000
Flip-Flops vs. Latches
Input/Output Behavior of Latches and Flipflops
Type When Inputs are Sampled When Outputs are Valid
unclocked always propagation delay from
latch input change
level clock high propagation delay from
sensitive (Tsu, Th around input change
latch falling clock edge)
positive edge clock lo-to-hi transition propagation delay from
flipflop (Tsu, Th around rising edge of clock
rising clock edge)
negative edge clock hi-to-lo transition propagation delay from
flipflop (Tsu, Th around falling edge of clock
falling clock edge)
master/slave clock hi-to-lo transition propagation delay from
flipflop (Tsu, Th around falling edge of clock
falling clock edge)

14. ELEC 256 / Saif Zahir
UBC / 2000
Flip-Flops: Typical Timing Specifications
74LS74 Positive
Edge Triggered
D Flipflop
• Setup time
• Hold time
• Minimum clock width
• Propagation delays
(low to high, high to low,
max and typical)
All measurements are made from the clocking event
that is, the rising edge of the clock
D
Clk
Q
Tsu
20
ns
Th
5
ns
Tw
25
ns
Tplh
25 ns
13 ns
Tsu
20
ns
Th
5
ns
Tphl
40 ns
25 ns

15. ELEC 256 / Saif Zahir
UBC / 2000
Latches: Typical Timing Specifications
74LS76
Transparent
Latch
• Setup time
• Hold time
• Minimum Clock Width
• Propagation Delays:
high to low, low to high,
maximum, typical
data to output
clock to output
Measurements from falling clock edge
or rising or falling data edge
Tsu
20
ns
Th
5
ns
Tsu
20
ns
Th
5
ns
Tw
20
ns
Tplh
C » Q
27 ns
15 ns
Tphl
C » Q
25 ns
14 ns
Tplh
D » Q
27 ns
15 ns
Tphl
D » Q
16 ns
7 ns
D
Clk
Q

16. ELEC 256 / Saif Zahir
UBC / 2000
Designing Latches
RS Latch
Truth Table:
Next State = F(S, R, Current State)
Derived K-Map:
Characteristic Equation:
q(t+D)=s(t)+R(t)q(t)
or
q+=s + Rq
R
SR
00 01 11 10
0 0 X 1
1 0 X 1
0
1
Q(t)
S
q
Q
R
S
q
q
R
S
Compare to previous NOR implementation

17. ELEC 256 / Saif Zahir
UBC / 2000
The JK Latch
The JK latch eliminates the forbidden state of the RS latch
Basic principle:
use output feedback to guarantee
that R=S=1 never occurs
J=K=1 yields toggle (q+ = Q)
Characteristic Equation:
Q+ = Q K + Q J
R-S
latch
K
J S
R
Q
Q
Q
Q
J
K
D
C
Q
enb
D-Latch

18. ELEC 256 / Saif Zahir
UBC / 2000
JK Latches
q SR SR SR SR
00 01 10 11
0 0 0 1 x
1 1 0 1 x
Q Q 0 1 x
PS NS (q+, Q+)
Simplified State-Tables
q JK JK JK JK
00 01 10 11
0 0 0 1 1
1 1 0 1 0
Q Q 0 1 Q
PS NS (q+, Q+)
JK=01 , 11
JK=10 , 11
JK=00 , 10
JK=00, 01
Q=1 Q=0
J K Q+
0 0 Q
0 1 0
1 0 1
1 1 Q

19. ELEC 256 / Saif Zahir
UBC / 2000
From JK Latch to JK Flip-Flop
JK Latch: Race Condition
J
K
Q
Q
100
Set Reset Toggle
Race Condition
• Ideally, the Latch should toggle only once when JK=11.
• Because of latch transparency, race conditions cause continuous toggrling.
• Toggle Correctness: Single State change per clocking event
• Solution: Master-Slave Flipflop

20. ELEC 256 / Saif Zahir
UBC / 2000
Master-Slave JK Flip-Flop
Correct Toggle
Operation
Master Stage Slave Stage
Sample inputs while clock high Sample inputs while clock low
J
R-S
Latch
R-S
Latch
K
R
S
Clk
Q
Q
P
P
R
S
Q
Q
Q
Q
Master
outputs
Slave
outputs
Set Reset Toggle
1's
Catch 100
J
K
Clk
P
P
Q
Q
Break feedback path, by dividing operation in two time periods
(clock-high and clock-low)

21. ELEC 256 / Saif Zahir
UBC / 2000
The Toggle (T) FlipFlop
State table
T Q Q+
0 0 0
0 1 1
1 0 1
1 1 0
T Q
C
T
flipflop
JK
flipflop
T J
K
C
Q
T-FF can be realized using a JK-FF
Verification: J=K=T
T Q+
0 Q
1 Q
or
T J K Q+
0 0 0 q
1 1 1 Q
q+ = tQ+Tq
D
flipflop
T
D
C
Q
T-FF can be realized using a D-FF

22. ELEC 256 / Saif Zahir
UBC / 2000
Edge-Triggered FlipFlops
Characteristic equation
Q+ = D
Q
Q
D
Clk=1
R
S
0
0
D
D
D
Holds D when
clock goes low
Holds D when
clock goes low
Negative edge-triggered D flipflop
• Flipflop state changes right after the falling edge of the clock
• 4-5 gate delays (longer than latches)
• Setup and Hold times are necessary for correct operation
Example:
D
Clk
Q

23. ELEC 256 / Saif Zahir
UBC / 2000
Edge-Triggered D FlipFlopk
Step-by-step analysis
Q
Q
D
Clk=0
R
S
D
D
D
D
D
D
When clock goes from
high-to-low data is latched
Q
Q
D'
Clk=0
R
S
D
D
D
D
D' ° D
0
0
1
2
3
4
5
6
When clock is low data is held

24. ELEC 256 / Saif Zahir
UBC / 2000
Positive and Negative Edge Triggered FlipFlops
Positive Edge Triggered
Inputs sampled on rising edge
Outputs change after rising edge
Negative Edge Triggered
Inputs sampled on falling edge
Outputs change after falling edge
Positive edge-
triggered FF
Negative edge-
triggered FF
D
Clk
Qpos
Qpos
Qneg
Qneg
100
Timing Diagram

25. ELEC 256 / Saif Zahir
UBC / 2000
Comparison
R-S Clocked Latch:
used as storage element in narrow width clocked systems
its use is not recommended!
however, fundamental building block of other flipflop types
J-K Flipflop:
versatile building block
can be used to implement D and T FFs
usually requires least amount of logic to implement ƒ(In,Q,Q+)
but has two inputs with increased wiring complexity
because of 1's catching, never use master/slave J-K FFs
Use edge-triggered varieties
D Flipflop:
minimizes wires, much preferred in VLSI technologies
simplest design technique
best choice for storage registers
T Flipflops:
don't really exist, constructed from J-K FFs
usually best choice for implementing counters
Asynchronous Preset and Clear inputs are highly desirable!

26. ELEC 256 / Saif Zahir
UBC / 2000
FlipFlop Excitation Tables
Useful Design Tool:
For each state-transition, the excitation table lists the required input combination(s)
D Q+
0 0
1 1
D Q
C
D
flipflop
q+ = d
T Q
C
T
flipflop
q+ = tQ+Tq
Q Q+ D
0 0 0
0 1 1
1 0 0
1 1 1
Excitation Table
Q Q+ T
0 0 0
0 1 1
1 0 1
1 1 0
1. D FlipFlop
2. T FlipFlop
Transition Table
T Q+
0 q
1 Q
Excitation Table
Transition Table

27. ELEC 256 / Saif Zahir
UBC / 2000
FlipFlop Excitation Tables
q+ = s + Rq
Q Q+ R S
0 0 X 0
0 1 0 1
1 0 1 0
1 1 0 X
1. SR FlipFlop
R Q
Clk SR
flipflop
S
Transition Table Excitation Table
R S Q+
0 0 Q
0 1 1
1 0 0
1 1 forbid
q+ = jQ + Kq
Q Q+ J K
0 0 0 X
0 1 1 X
1 0 X 1
1 1 X 0
1. JK FlipFlop
J Q
Clk JK
flipflop
K
Transition Table Excitation Table
R S Q+
0 0 q
0 1 1
1 0 0
1 1 Q
Q=0 Q=1
JK= 10, 11
JK= 01, 11
JK=00,01 JK=00,10
Q=0 Q=1
RS= 01
RS=10
RS=00,10 RS=00,01

28. ELEC 256 / Saif Zahir
UBC / 2000
Conversion Between FlipFlop Types
Procedure uses excitation tables
Method: to realize a type A flipflop using a type B flipflop:
1. Start with the K-map or state-table for the A-flipflop.
2. Express B-flipflop inputs as a function of the inputs and present state of
A-flipflop such that the required state transitions of A-flipflop are reallized.
x
y
Q
Type B
x
y
Q
g
h
CL
CL
Type A
1. Find Q+ = f(g,h,Q) for type A (using type A state-table)
2. Compute x = f1(g,h,Q) and y=f2(g,h,Q) to realize Q+.

29. ELEC 256 / Saif Zahir
UBC / 2000
Conversion Between FlipFlop Types
Example: Use JK-FF to realize D-FF
1) Start transition table for D-FF
2) Create K-maps to express J and K as functions of inputs (D, Q)
3) Fill in K-maps with appropriate values for J and K
to cause the same state transition as in the D-FF transition table
D
0
1
0
1
T
0
1
1
0
Q+
0
1
0
1
Q
0
0
1
1
S
0
1
0
X
R
X
0
1
0
K
X
X
1
0
J
0
1
X
X
D
X X
1 0
K = D
0 1
0
1
Q
D
0 1
X X
J = D
0 1
0
1
Q
State-Table
D Q Q+ J K
0 0 0 0 X
0 1 0 X 1
1 0 1 1 X
1 1 1 X 0
e.g.
when D=Q=0, then Q+= 0
the same transition Q-->Q+
is realize with J=0, K=X

30. ELEC 256 / Saif Zahir
UBC / 2000
Conversion Between FlipFlops
Another Example: Implement JK-FF using a D-FF
J K Q Q+ D T
0 0 0 0 0 0
0 0 1 1 1 0
0 1 0 0 0 0
0 1 1 0 0 1
1 0 0 1 1 1
1 0 1 1 1 0
1 1 0 1 1 1
1 1 1 0 0 1
0 0 1 1
0 1 1 0
00 01 11 10
J
K
JK
Q
0
1
t= jQ + kq
0 0 1 1
1 0 0 1
00 01 11 10
J
K
JK
Q
0
1
d= jQ + Kq
J
K
D
C
Q
Clk
DFF
J
K
T
C
Q
Clk
T-FF

31. ELEC 256 / Saif Zahir
UBC / 2000
Asynchronous Inputs
PRESET and CLEAR:
asynchronous, level-sensitive inputs
used to initialize a flipflop.
D
C
S
R
Q
Q
0
1
0
1
0
1 Q
Clk
SET
CLR
T
Q
T
SET
CLR
Clk
200 400
Clk
T Q
CLEAR
PRESET
PRESET, CLEAR: active low inputs
PRESET = 0 --> Q = 1
CLEAR = 0 --> Q = 0
LogicWorks Simulation

32. ELEC 256 / Saif Zahir
UBC / 2000
In
Q0
Q1
Clk
100
Proper Cascading of Flipflops
Correct Operation,
assuming positive
edge triggered FF
IN
CLK
Q0 Q1
D
C
Q
Q
D
C
Q
Q
FF0 FF1
Serial connection of positive edge-trigerred flipflops
1. on rising efge of CLK, FF1 reads Q0, and FF0 reads IN
2. during clock period FF1 performs Q1 <-- Q0, and FF0 performs Q0 <-- IN
Shift-register